Modal Shapes

A modal shape represents the motion of the guitar in a normal mode of vibration. Optical methods give the best spatial resolution of a given operational deflection shape (ODS), which in many cases closely resembles a normal mode. Optical methods include holographic interferometry, speckle-pattern interferometry, and scanning laser vibrometry.

Another technique for obtaining modal shapes, called experimental modal testing, excites the guitar body with a force hammer and uses an accelerometer to

3 Guitars and Lutes 29

Fig. 3.12 Holographic interferograms of a classical guitar top plate at several resonances. Resonance frequencies and Q-values (a measure of the sharpness of the resonance) are given (Richardson and Roberts 1985)

sense its motion. The force hammer is moved from point to point in a grid, and a frequency response function (FRF) determined for each point of excitation. The resulting FRFs are processed by a computer and the modal shape is determined by use of a curve-fitting program.

3.3 String Forces

A player can alter the tone of a guitar by adjusting the angle through which the string is plucked. Not only do forces parallel and perpendicular to the bridge excite different sets of resonances, but they result in tones that have different decay rates, as shown in Fig. 3.13. When the string is plucked perpendicular to the top plate, a strong but rapidly decaying tone is obtained. When the string is plucked parallel to the plate, on the other hand, a weaker but longer tone results. Thus, a guitar tone can be regarded as having a compound decay rate, as shown in Fig. 3.13 (bottom). The spectra of the initial and final parts of the tone vary substantially, as do the decay rates.

Classical guitarists use primarily two strokes, called apoyando and tirando (sometimes called the rest and free strokes). The fingernail acts as sort of a ramp, converting some of the horizontal motion of the finger into vertical motion of the string, as shown in Fig. 3.14. Although the apoyando stroke tends to induce slightly more vertical string motion, there is little difference between the two strokes in this

30 T.D. Rossing and G. Caldersmith

Fig. 3.13 Decay rates of guitar tone for different plucking directions (Jansson 1983)

Fig. 3.14 Finger motion and resulting string motion of apoyando and tirando strokes. In the apoyando stroke, the finger comes to rest on an adjacent string; in the tirando stroke, it rises enough to clear it (Taylor 1978)

regard. However, the player can change the balance between horizontal and vertical string motion by varying the angle of the fingertip (Taylor 1978).

Guitar Sound Radiation

Sound radiation from a guitar, like most musical instruments, varies with direction and frequency. Even with sinusoidal excitation at a single point (such as the bridge), the radiated sound field is complicated because several different modes of vibration with different patterns of radiation may be excited at the same time. Figure 3.15

Guitars and Lutes 31

Fig. 3.15 Mechanical frequency response and sound spectrum one meter in front of a Martin D-28 steel-string guitar driven by a sinusoidal force of 0.15 N applied to the treble side of the bridge. The solid curve is the sound spectrum; the dashed curve is acceleration at the driving point

Fig. 3.16 Sound radiation patterns at four resonance frequencies in a Martin D-28 folk guitar (compare with Fig. 3.7 which show the corresponding modal shapes) (Popp and Rossing 1986)

shows the sound spectrum one meter in front of a Martin D-28 folk guitar in an anechoic room when a sinusoidal force of 0.15 N is applied to the treble side of the bridge. Also shown is the mechanical frequency response curve (acceleration level versus frequency). Note that most of the mechanical resonances result in peaks in the radiated sound, but that the strong resonances around 376 and 436 Hz (which represent “seesaw” motion; see Fig. 3.11) do not radiate strongly in this direction. The mode at 102 Hz radiates efficiently through the sound hole.

Figure 3.16 shows polar sound radiation patterns in an anechoic room for the modes at 102, 204, 376, and 436 Hz. The modes at 102 and 204 Hz radiate quite efficiently in all directions, as would be expected in view of the mode shapes (see Fig. 3.7). Radiation at 376 Hz, however, shows a dipole character, and at 436 Hz a strong quadruple character is apparent, as expected from Fig. 3.7 (Popp and Rossing 1986).

32 T.D. Rossing and G. Caldersmith

Fig. 3.17 Comparison of the sound level of the fundamentals of played notes (bars) to the guitar frequency response function (solid curve) with its level adjusted for a good fit. A graph of the rate of sound decay (dB/s) versus frequency similarly follows the frequency response curve (Caldersmith and Jansson 1980)

The output spectrum of a guitar may be calculated by multiplying the bridge force spectrum by the frequency response function of the guitar body. This is greatly complicated, however, by the rapid change in the force spectrum with the time after the pluck (see Fig.3.13). Caldersmith and Jansson (1980) measured the initial sound level and the rate of sound decay for played notes on guitars of high and medium quality. They found that both the initial sound level and the rate of decay replicate the frequency response curve of a guitar, as shown in Fig. 3.17. At strong resonances, however, the initial levels are slightly lower, and the levels decay faster than predicted by the frequency response curves.

3.5 Quality

Rating the sound quality of classical guitars and how the quality depends on design and construction details have been studied by several investigators. According to Jansson (2002), most guitar players feel that tonal strength or carrying power is the most important single quality criterion, with tone length and timbre being the second most important. In the previous section, we mentioned how the initial sound level and rate of sound decay depends upon the resonances of a guitar body.

Tones from recorded music were analyzed in the form of long time average spectra (LTAS), and it was found that better guitars have a higher level up to 3,000 Hz. Comparing two guitars, it was found that the less good guitars tended to have a lower level below 2,000 Hz and above 400 Hz (Jansson 2002).

The Guitars and The Oud

Some extensive listening tests were conducted at the Physikalisch-Technische Bundesanstalt in Germany to try to correlate quality in guitars to their measured frequency response (Meyer 1983). Some of the features that correlated best with high quality were:

1. The peak level of the third resonance (around 400 Hz);
2. The amount by which this resonance stands above the resonance curve level; 3. The sharpness (Q value) of this resonance;
4. The average level of one-third-octave bands in the range 80–125 Hz;
5. The average level of one-third-octave bands in the range 250–400 Hz;
6. The average level of one-third-octave bands in the range 315–5,005 Hz;
7. The average level of one-third-octave bands in the range 80–1,000 Hz;
8. The peak level of the second resonance (around 200 Hz).

3.5.1 Influence of Design and Construction

Meyer found that using fewer struts, varying their spacing, adding transverse bracing and reducing the size of the bridge, to have desirable effects (Meyer 1983). He experimented with several different bridge shapes and found that a bridge without “wings” gave the best result.

Jansson (2002) found the following order of importance for different parts in determining quality:

1. Bridge
2. Top plate thickness 3. Cross bars or struts.

So-called “frame” guitar designs have a rigid waist bar to inhibit leakage of vibrational energy from the lower bout to the upper bout and other parts of the guitar.

 The Bridge

The bridge has a marked stiffening effect on the top plane, and thus affects the vibrations. For a heavy bridge the frequency of the first top plate resonance may decrease, the mass giving a larger contribution than the stiffness increase. Hand- made Spanish bridges tend to be considerably lighter and less rigid than factory- made bridges. For low frequencies the mass increase may dominate, but at higher frequencies the stiffening effect dominates (Jansson 2002).

3.5.3 Thickness of the Top Plate and Braces

Richardson and Roberts (1985) studied the influence of top plate and strut thickness with finite-element modeling using a computer. At the start, the plate thickness

34 T.D. Rossing and G. Caldersmith

Fig. 3.18 Lattice bracing of a guitar top plate used by Australian luthier Greg Smallman. Struts are typically of carbon-fiber- epoxy, thickest at the bridge and tapering away from the bridge in all directions (Caldersmith and Williams 1986)

was 2.9 mm, and the struts were 14 mm high and 5 mm wide. Their calculations showed that the cross struts gave a large influence at least for the low resonances. A reduction in strut height also results in a large influence on the resonance frequen- cies. Reduction in top plate thickness, especially thinning along the edge, has the greatest effect of all.

Richardson and his students have also found that reducing the effective mass has a great effect on radiation of high-frequency sound, even more than tuning the mode frequencies (Richardson 1998). The effective mass is difficult to control, however, after the choice of materials and general design has been made. Of primary importance is the effective mass of the fundamental sound board mode.

Australian luthier Greg Smallman, who builds guitars for John Williams, has enjoyed considerable success by using lightweight top plates supported by a lattice of braces, the heights of which are tapered away from the bridge in all directions, as shown in Fig. 3.18. Smallman generally uses struts of carbon-fiber-epoxy expoxied to balsa wood (typically 3 mm wide and 8 mm high at their tallest point) in order to achieve high stiffness-to-mass ratio and hence high-resonance frequencies or “lightness” (Caldersmith and Williams 1986).

3.5.4 Asymmetrical and Radial Bracing

Although many classical guitars are symmetrical around their center plane, a number of luthiers (e.g., Hauser in Germany and Ramirez in Spain, Schneider and Eban in the United States) have had considerable success by introducing varying degrees of asymmetry into their designs. Most asymmetric guitars have shorter but thicker struts on the treble side, thus making the plate stiffer. Three such top plate designs are shown in Fig. 3.19.

3 Guitars and Lutes 35

Fig. 3.19 Examples of asymmetric top plates: (a) Ramirez (Spain); (b) Fleta (Spain); (c) Eban (United States)

Fig. 3.20 Holographic interferograms showing modal shapes of two low-frequency modes at 101 and 304 Hz in a radially braced classical guitar (Rossing and Eban 1999)

The very asymmetric design in Fig. 3.19c was proposed by Kasha (1974) and developed by luthiers Richard Schneider, Gila Eban, and others. It has a split asymmetric bridge (outlined by the dashed line) and closely spaced struts of varying length. A waist bar (WB) bridges the two long struts and the sound hole liner.

Despite its asymmetry the vibrational modal shapes, at least at low frequency, are quite similar to other good classical guitars, as shown in the holographic interfero- grams in Fig. 3.20. The particular guitar in this modal study had a one-piece bridge and radial bracing in the back plate as well as the top plate. Other luthiers have had considerable success with radial bracing. Australian luthier Simon Marty uses a radial bracing of balsa or cedar reinforced with carbon fiber. Trevor Gore has had success using falcate bracing with curved braces of balsa and carbon fiber.

3.6 A Family of Scaled Guitars

Members of guitar ensembles (trios, quartets) generally play instruments of similar design, but Australian physicist/luthier Graham Caldersmith has created a new family of guitars especially designed for ensemble performance. (Actually, he has created two such families: one of classical guitars and one of steel-string folk guitars). His classical guitar family, including a treble guitar, a baritone guitar,

36 T.D. Rossing and G. Caldersmith

and a bass guitar in addition to the conventional guitar – which becomes the tenor of the family – has been played and recorded extensively by the Australian quartet Guitar Trek (Caldersmith 1995).

Caldersmith’s guitar families include carefully scaled instruments, the tunings and resonances of which are translated up and down by musical fourths and fifths, in much the same way as the Hutchins–Schelleng violin octet (see Chap. 18). Calder- smith’s bass guitar is a four-string instrument tuned the same as the string bass and the electric bass (E1, A1, D2, G2), an octave below the four lowest strings of the standard guitar. The baritone is a six-string instrument tuned a musical fifth below the standard, while the treble is tuned a musical fourth above the standard, being then an octave above the baritone. Caldersmith uses an internal frame, but a graded rectangular lattice instead of the diagonal lattice (see Fig. 3.21). The Australian Guitar Quartet is shown in Fig. 3.22.

Fig. 3.21 Caldersmith guitar with internal frame and rectangular lattice

Fig. 3.22 The Australian Guitar Quartet play on scaled guitars: bass and baritone by Graham Caldersmith, standard and treble by Greg Smallman and Eugene Philp

Guitars and Lutes 37

3.7 Synthetic Materials

Traditionally guitars have top plates of spruce or redwood with backs and ribs of rosewood or some comparable hardwood. Partly because traditional woods are sometimes in short supply, luthiers have experimented with a variety of other woods, such as cedar, pine, mahogany, ash, elder, and maple. Bowls of fiberglass, used to replace the wooden back and sides of guitars, were developed by the Kaman company in 1966; their Ovation guitars have become popular, partly because of their great durability.

One of the first successful attempts to build a guitar mostly of synthetic materials was described by Haines et al. (1975). The body of this instrument, built to the dimensions of a Martin folk guitar, used composite sandwich plates with graphite- epoxy facings around a cardboard core. In listening tests, the guitar of synthetic material was judged equal to the wood standard for playing scales, but inferior for playing chords. In France, Charles Besnainou and his colleagues have constructed lutes, violins, violas, cellos, double basses, and harpsichords, as well as guitars, using synthetic materials (Besnainou 1995).

3.8 Other Families of Guitars

Most of our discussion has been centered on classical guitars, with occasional compar- ison to the steel-string American folk (flat top) guitar. There are several other types of acoustic guitars in use throughout the world, including flamenco, archtop, 12-string, jazz, resonator, etc. Portuguese guitars will be discussed in Chap. 4. Some Asian plucked string instruments of the lute family will be discussed in Chap. 11.

The gypsy guitar, known in France as the manouche guitar, gained popularity in the late 1920s. Played by Django Reinhardt throughout his career, the instrument has seen a revival in interest. The community of gypsy jazz players today is a small, but growing one, and the original Selmer–Maccaferri guitars are highly valued and widely copied. Its low-gauge strings offer its player a brighter, more metallic tone, with an ease for creating a very distinct vibrato (Lee et al. 2007).